Product rule
The derivative of u times v is u'v + uv'. You cannot just multiply the two derivatives.
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Analogy
A rectangle with growing sides: area changes because the width grows and because the height grows.
Example
u = x^2, u' = 2x. v = e^x, v' = e^x.u'v + uv' = 2x e^x + x^2 e^x.Factor: x e^x (2 + x).
In practice
Use it whenever two non-constant factors, each with x in them, are multiplied.
Practice
Flashcards
- State the product rule.(uv)' = u'v + uv'.
Interview questions
- Differentiate f(x) = x^2 sin(3x) and state which rules you used.
- Product rule with u = x^2 and v = sin(3x)
- Power rule: u' = 2x
- Chain rule: v' = 3 cos(3x)
- Result 2x sin(3x) + 3x^2 cos(3x)
Questions people ask
Why is the derivative of a product not the product of derivatives?
Both factors change at once, and each change affects the total. Take x times x: the product of derivatives is 1, but the true derivative of x squared is 2x. The product rule adds both contributions: u prime v plus u v prime.
How do I remember the product rule?
Say it in words: derivative of the first times the second, plus the first times derivative of the second. Differentiate one factor at a time while leaving the other alone, then add the results. It works for any two differentiable functions.